In probability theory and statistics , the U-quadratic distribution is a continuous probability distribution defined by a unique convex quadratic function with lower limit a and upper limit b .
U-quadratic
Probability density function
Parameters
a
:
a
∈
(
−
∞
,
∞
)
{\displaystyle a:~a\in (-\infty ,\infty )}
b
:
b
∈
(
a
,
∞
)
{\displaystyle b:~b\in (a,\infty )}
or
α
:
α
∈
(
0
,
∞
)
{\displaystyle \alpha :~\alpha \in (0,\infty )}
β
:
β
∈
(
−
∞
,
∞
)
,
{\displaystyle \beta :~\beta \in (-\infty ,\infty ),}
Support
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]\!}
PDF
α
(
x
−
β
)
2
{\displaystyle \alpha \left(x-\beta \right)^{2}}
CDF
α
3
(
(
x
−
β
)
3
+
(
β
−
a
)
3
)
{\displaystyle {\alpha \over 3}\left((x-\beta )^{3}+(\beta -a)^{3}\right)}
Mean
a
+
b
2
{\displaystyle {a+b \over 2}}
Median
a
+
b
2
{\displaystyle {a+b \over 2}}
Mode
a
and
b
{\displaystyle a{\text{ and }}b}
Variance
3
20
(
b
−
a
)
2
{\displaystyle {3 \over 20}(b-a)^{2}}
Skewness
0
{\displaystyle 0}
Excess kurtosis
3
112
(
b
−
a
)
4
{\displaystyle {3 \over 112}(b-a)^{4}}
Entropy
TBD MGF
See text CF
See text
f
(
x
|
a
,
b
,
α
,
β
)
=
α
(
x
−
β
)
2
,
for
x
∈
[
a
,
b
]
.
{\displaystyle f(x|a,b,\alpha ,\beta )=\alpha \left(x-\beta \right)^{2},\quad {\text{for }}x\in [a,b].}
Parameter relations
edit
This distribution has effectively only two parameters a , b , as the other two are explicit functions of the support defined by the former two parameters:
β
=
b
+
a
2
{\displaystyle \beta ={b+a \over 2}}
(gravitational balance center, offset), and
α
=
12
(
b
−
a
)
3
{\displaystyle \alpha ={12 \over \left(b-a\right)^{3}}}
(vertical scale).
Related distributions
edit
One can introduce a vertically inverted (
∩
{\displaystyle \cap }
)-quadratic distribution in analogous fashion.
Applications
edit
This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution – e.g., beta distribution and gamma distribution .
Moment generating function
edit
M
X
(
t
)
=
−
3
(
e
a
t
(
4
+
(
a
2
+
2
a
(
−
2
+
b
)
+
b
2
)
t
)
−
e
b
t
(
4
+
(
−
4
b
+
(
a
+
b
)
2
)
t
)
)
(
a
−
b
)
3
t
2
{\displaystyle M_{X}(t)={-3\left(e^{at}(4+(a^{2}+2a(-2+b)+b^{2})t)-e^{bt}(4+(-4b+(a+b)^{2})t)\right) \over (a-b)^{3}t^{2}}}
Characteristic function
edit
ϕ
X
(
t
)
=
3
i
(
e
i
a
t
e
i
b
t
(
4
i
−
(
−
4
b
+
(
a
+
b
)
2
)
t
)
)
(
a
−
b
)
3
t
2
{\displaystyle \phi _{X}(t)={3i\left(e^{iate^{ibt}}(4i-(-4b+(a+b)^{2})t)\right) \over (a-b)^{3}t^{2}}}